pca matlab代码调用,[转载]pca的MATLAB源代码（princomp）

蔚落 2023-01-16 14:26 146阅读 0赞

function \[coeff, score, latent,

tsquare\] = princomp(x,econFlag)

%PRINCOMP Principal Components

Analysis.

% COEFF =

PRINCOMP(X) performs principal components analysis on the

N-by-P

% data

matrix X, and returns the principal component coefficients,

also

% known as

loadings. Rows of X correspond to

observations, columns to

% variables. COEFF is a P-by-P matrix, each

column containing coefficients

% for one

principal component. The columns are in

order of decreasing

% component

variance.

% PRINCOMP

centers X by subtracting off column means, but does not

% rescale the

columns of X. To perform PCA with

standardized variables,

% i.e., based on

correlations, use PRINCOMP(ZSCORE(X)). To

perform PCA

% directly on a

covariance or correlation matrix, use PCACOV.

% \[COEFF, SCORE\]

= PRINCOMP(X) returns the principal component scores,

% i.e., the

representation of X in the principal component

space. Rows

% of SCORE

correspond to observations, columns to components.

% \[COEFF, SCORE,

LATENT\] = PRINCOMP(X) returns the principal component

% variances,

i.e., the eigenvalues of the covariance matrix of X, in

% LATENT.

% \[COEFF, SCORE,

LATENT, TSQUARED\] = PRINCOMP(X) returns Hotelling's

% T-squared

statistic for each observation in X.

% When N

<= P, SCORE(:,N:P) and LATENT(N:P) are necessarily

zero, and the

% columns of

COEFF(:,N:P) define directions that are orthogonal to X.

% \[...\] =

PRINCOMP(X,'econ') returns only the elements of LATENT that

are

% not necessarily

zero, i.e., when N <= P, only the first N-1,

and the

% corresponding

columns of COEFF and SCORE. This can be

significantly

% faster when P

>> N.

% See also

BARTTEST, BIPLOT, CANONCORR, FACTORAN, PCACOV, PCARES,

ROTATEFACTORS.

% References:

% \[1\] Jackson, J.E., A User's Guide to Principal Components,

% Wiley, 1988.

% \[2\] Jolliffe, I.T. Principal Component Analysis, 2nd ed.,

% Springer, 2002.

% \[3\] Krzanowski, W.J., Principles of Multivariate Analysis,

% Oxford University Press, 1988.

% \[4\] Seber, G.A.F., Multivariate Observations, Wiley, 1984.

% Copyright

1993-2009 The MathWorks, Inc.

% $Revision:

2.9.2.11 $ $Date: 2009/01/30 14:50:55

% When X has more variables than

observations, the default behavior is to

% return all the pc's, even those that

have zero variance. When econFlag

% is 'econ', those will not be

returned.

if nargin < 2, econFlag

= 0; end

\[n,p\] = size(x);

if isempty(x)

pOrZero = ~isequal(econFlag, 'econ') \* p;

coeff = zeros(p,pOrZero); coeff(1:p+1:end) = 1;

score = zeros(n,pOrZero);

latent = zeros(pOrZero,1);

tsquare = zeros(n,1);

return

end

% Center X by subtracting off column

means

x0 = bsxfun(@minus,x,mean(x,1));

if nargout < 2

% The principal component coefficients are the eigenvectors of

% S = X0'\*X0./(n-1), but computed using SVD.

if n

>= p &&

(isequal(econFlag,0) || isequal(econFlag,'econ'))

\[coeff,ignore\] = eig(x0'\*x0);

coeff = fliplr(coeff);

else

\[ignore,ignore,coeff\] = svd(x0,econFlag);

end

% When econFlag is 'econ', only (n-1) components should be

returned.

% See comment below.

if (n <= p) &&

isequal(econFlag, 'econ')

coeff(:,n) = \[\];

end

else

r = min(n-1,p); % max possible rank of X0

% The principal component coefficients are the eigenvectors of

% S = X0'\*X0./(n-1), but computed using SVD.

\[U,sigma,coeff\] = svd(x0,econFlag); % put in 1/sqrt(n-1) later

% Project X0 onto the principal component axes to get the

scores.

if n == 1 % sigma might have only 1 row

sigma = sigma(1);

else

sigma = diag(sigma);

end

score = bsxfun(@times,U,sigma'); % == x0\*coeff

sigma = sigma ./ sqrt(n-1);

% When X has at least as many variables as observations,

eigenvalues

% n:p of S are exactly zero.

if n <= p

% When econFlag is 'econ', nothing corresponding to the zero

% eigenvalues should be returned. svd(,'econ') won't have

% returned anything corresponding to components (n+1):p, so we

% just have to cut off the n-th component.

if isequal(econFlag, 'econ')

sigma(n,:) = \[\]; % make sure this shrinks as a column

coeff(:,n) = \[\];

score(:,n) = \[\];

% Otherwise, set those eigenvalues and the corresponding scores

% exactly zero. svd(,0) won't have

returned columns of U

% corresponding to components (n+1):p, need to fill those out.

else

sigma(n:p,1) = 0; % make sure this extends as a column

score(:,n:p) = 0;

end

% The variances of the pc's are the eigenvalues of S =

X0'\*X0./(n-1).

latent = sigma.^2;

% Hotelling's T-squared statistic is the sum of squares of the

% standardized scores, i.e., Mahalanobis

distances. When X appears to

% have column rank < r, ignore components that are

orthogonal to the

% data.

if nargout == 4

if n > 1

q = sum(sigma > max(n,p).\*eps(sigma(1)));

if q < r

warning('stats:princomp:colRankDefX', ...

\['Columns of X are linearly dependent to within machine

precision.n' ...

'Using only the first %d components to compute

TSQUARED.'\],q);

end

else

q = 0;

end

tsquare = (n-1) .\* sum(U(:,1:q).^2,2); % ==

sum((score\*diag(1./sigma)).^2,2)

end

%% test for princomp(Principal

Component Analysis)

% 关于matlab中princomp的使用说明讲解小例子 by

faruto

能看懂本程序及相关注释讲解的前提是您对PCA有一定的了解~O(∩\_∩)O

% 2009.10.27

clear;

clc

%% load cities data

load cities

% whos

% Name Size Bytes Class

% categories 9x14 252 char

array

% names 329x43 28294 char

array

% ratings 329x9 23688 double array

%% box plot for ratings data

% To get a quick impression of the

ratings data, make a box plot

figure;

boxplot(ratings,'orientation','horizontal','labels',categories);

grid on;

print -djpeg 1;

%% pre-process

stdr = std(ratings);

sr = ratings./repmat(stdr,329,1);

%% use princomp

\[coef,score,latent,t2\] =

princomp(sr);

%% 输出参数讲解

% coef:9\*9

主成分系数:即原始数据线性组合生成主成分数据中每一维数据前面的系数.

% coef的每一列代表一个新生成的主成分的系数.

% 比如你想取出前三个主成分的系数,则如下可实现:pca3 =

coef(:,1:3);

% score:329\*9

% 字面理解:主成分得分

% 即原始数据在新生成的主成分空间里的坐标值.

% latent:9\*1

% 一个列向量,由sr的协方差矩阵的特征值组成.

% 即 latent =

sort(eig(cov(sr)),'descend');

% 测试如下:

% sort(eig(cov(sr)),'descend') =

% 3.4083

% 1.2140

% 1.1415

% 0.9209

% 0.7533

% 0.6306

% 0.4930

% 0.3180

% 0.1204

% latent =

% 3.4083

% 1.2140

% 1.1415

% 0.9209

% 0.7533

% 0.6306

% 0.4930

% 0.3180

% 0.1204

% t2:329\*1

% 一中多元统计距离,记录的是每一个观察量到中心的距离

%% 如何提取主成分,达到降为的目的

% 通过latent,可以知道提取前几个主成分就可以了.

figure;

percent\_explained =

100\*latent/sum(latent);

pareto(percent\_explained);

xlabel('Principal Component');

ylabel('Variance Explained (%)');

print -djpeg 2;

% 图中的线表示的累积变量解释程度.

% 通过看图可以看出前七个主成分可以表示出原始数据的90%.

所以在90%的意义下只需提取前七个主成分即可,进而达到主成分提取的目的.

%% Visualizing the Results

% 结果的可视化

figure;

biplot(coef(:,1:2),

'scores',score(:,1:2),...

'varlabels',categories);

axis(\[-.26 1 -.51 .51\]);

print -djpeg 3;

% 横坐标和纵坐标分别表示第一主成分和第二主成分

% 红色的点代表329个观察量,其坐标就是那个score

蓝色的向量的方向和长度表示了每个原始变量对新的主成分的贡献,其坐标就是那个coef.

==============本帖子的pdf版本

\[hide\]\[/hide\]

===================

引用:

回复 8\# 5342245 的帖子 设原始数据为X，先不做任何预处理。

\[coef,score,latent,t2\] =

princomp(X);

则那些参数的底层算法大体过程如下：

x0 = bsxfun(@minus,X,mean(X,1));

%x0为将X去均值后的数据。

\[coef,ignore\] = eig(x0'\*x0);

这就是coef的由来。 【当然最终的还有排序什么乱七八糟的。。】

scroe = x0\*coef %

这就是score的由来，就是一个简单的线性变换，将原来的X的坐标转换到主成分空间中的坐标。仅此而已

则模型为从原始数据出发：

score =

bsxfun(@minus,X,mean(X,1))\*coef;

逆变换：

X =

bsxfun(@plus,score\*inv(coef),mean(X,1))

以上这些你可以自己验证，看是否正确。

关于你的第三问。对于每一个主成分，就看coef的相应的列就能知道原始的变量那个对该主成分贡献大了啊。。

上面是没有预处理的。如果加了可逆的预处理。则原始数据亦可从预处理后的数据表示出。进而 bla bla....

===============这回够通俗易懂吧。。O(∩\_∩)O

PS：pca算法流程，你熟悉吗？只要知道那个算法过程。这些都不难理解啊。。

建议您看看书把pca算法流程再过一遍。。否则别人再怎么说也没用。。。

pca matlab代码调用,[转载]pca的MATLAB源代码（princomp）

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